Question

Let X and Y be two continuous random variables with joint probability density function

f(x,y) =

6x 0<y<1, 0<x<y,

0 otherwise.

a) Find the marginal density of Y .

b) Are X and Y independent?

c) Find the conditional density of X given Y = 1 /2

Answer #1

Let X and Y be two continuous random variables with joint
probability density function f(x,y) = xe^−x(y+1), 0 , 0< x <
∞,0 < y < ∞ otherwise
(a) Are X and Y independent or not? Why?
(b) Find the conditional density function of Y given X = 1.(

For continuous random variables X and Y with joint probability
density function. f(x,y) = xe−(x+y) when x > 0 and y
> 0 f(x,y) = 0 otherwise
a. Find the conditional density F xly (xly)
b. Find the marginal probability density function fX (x)
c. Find the marginal probability density function fY (y).
d. Explain if X and Y are independent

Let X and Y be jointly continuous random variables with joint
density function f(x, y) = c(y^2 − x^2 )e^(−2y) , −y ≤ x ≤ y, 0
< y < ∞.
(a) Find c so that f is a density function.
(b) Find the marginal densities of X and Y .
(c) Find the expected value of X

Let X and Y be two continuous random variables with joint
probability density function
?(?, ?) = { ? 2 + ?? 3 0 ≤ ? ≤ 1, 0 ≤ ? ≤ 2 0 ??ℎ??????
Find ?(? + ? ≥ 1). Sketch the surface in the ? − ? plane.

X and Y are continuous random variables. Their joint probability
distribution function is :
f(x,y) = 1/5(y+2) , 0 < y < 1, y-1 < x < y +1
= 0, otherwise
a) Find marginal density of Y, fy(y)
b) Calculate E[X | Y = 0]

Suppose that the joint probability density function of the
random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤
x ≤ 1, 0 ≤ y ≤ 1 0 otherwise.
(a) Sketch the region of non-zero probability density and show
that c = 3/ 2 .
(b) Find P(X + Y < 1), P(X + Y = 1) and P(X + Y > 1).
(c) Compute the marginal density function of X and Y...

The joint probability density function of two random variables
(X and Y) is given by fX,Y (x, y) = ( C √y (y ^(α+1)) exp {( −
y(2β+x ^2 ) )/2 } , x ∈ (−∞,∞), y ∈ [0,∞), 0 otherwise. (a) Find C.
(b) Find the marginal density of Y . What type of distribution does
Y follow? (c) Find the conditional density of X | Y . What type of
distribution is this?

X and Y are continuous random variables. Their joint probability
density function is given as f(x,y) = 1/5 (y+2) for 0<y<1 and
y-1<x<y+1. Calculate the conditional expectation
E(x/y=0).
Please show all the work and explain if the answer will be a
number or just y in a given range.

Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { cx2y, 0 < x2 < y < x for
x > 0 0, otherwise }. compute the marginal probability density
functions fX(x) and fY (y). Are the random variables X and Y
independent?.

Suppose X and Y are continuous random variables with joint
density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
(a). Compute the joint CDF F(x,y).
(b). Compute the marginal density for X and Y .
(c). Compute Cov(X,Y ). Are X and Y independent?

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